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+ 7.1 Category Extensions

+ 7.1.1 Free extension

Let us have some category \(C\) wth sums, products, and exponentials (i.e. a cartesian closed distributive category), and suppose we wish to extend it. One way to do this is to add some extra objects and some arrows.

To make a new category we throw in identities for the new arrows and then throw in all the paths formed from the combined set of arrows.

However we do more than that: we also throw in all sums, products and exponentials as well.

Each such path always has at least one of the new arrows in it.

Note: this is an informal description. The application should make it obvious!

+ 7.1.2 Application: Felix class

We will given an example:

  class New {
    type T = "T";
    type U = "U";
    fun f: T -> T;
    fun g: T -> U;
  }

This is a graph, which extends its context. That means it implies that there may functions of type

    int * T -> U

for example. Remember, the text defining a function is an encoding, equality is defined by the semantics, not the representation. When we say a function exists, we mean the semantics can be encoded, not that the programmer has actually done so.

We have needed quite a lot of machinery here to just if the use of the word class, which is just a synonym for the word category. What we write in a class isn't a category as such, but a graph that generates an extension to a category.

The implication is that the system throws in sums, products and exponentials for you: this is called structural typing. The functions in Felix are usually encoded with applicative and not compositional syntax.